Percutaneous transluminal coronary angioplasty involves percutaneous introduction of a balloon catheter into an obstructed coronary artery and then inflating the balloon within the obstruction to widen the luminal passage through the artery to improve blood flow through the artery. Among the techniques that have developed is the "kissing balloon" technique which is used to treat an obstruction in the region of a bifurcation of an artery. Difficulty may arise when attempting to perform a dilatation in one of the branches of the bifurcated artery. In such a situation, the inflation of the balloon dilatation catheter in one artery may cause closure of the other adjacent branch artery. In order to prevent that occurrence, the "kissing balloon" technique was developed. In that technique, two balloon dilatation catheters are used, side-by-side, one catheter extending into one branch of the bifurcation and the other catheter extending into the other branch of the bifurcation. If the dilatation of the stenosed branch causes the other branch artery to become constricted, the balloon in that other branch then can be inflated to prevent closure of that other branch. Typically, in such an arrangement, the proximal ends of the balloons are disposed within the common trunk artery and their proximal ends touch or "kiss". The difficulty with this arrangement, however, is that when both balloons are inflated, the effective diameter at their proximal, kissing, ends may be too large for the diameter of the common trunk artery, thus, risking injury to that common artery. It is among the general objects of the present invention to provide an improved catheter construction that reduces the risk of damage to the common artery.